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Chapter 6: Problem 36

Let \(x\) be a continuous random variable that is normally distributed with a mean of 65 and a standard deviation of 15 . Find the probability that \(x\) assumes a value a. less than 45 b. greater than 79 c. greater than 54 d. less than 70

Short Answer

Expert verified
a. 0.0925, b. 0.1762, c. 0.7673, d. 0.6293

Step by step solution

01

Definition of Z-score

The Z-score of a data point is defined as \(Z = \frac{x - \mu}{\sigma}\) where \(x\) is the value of the data point, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation. We'll use this formula to convert the given random variable values to their respective Z-scores.
02

Using Z-score formula for a.

We have \(x = 45\), \(\mu = 65\), and \(\sigma = 15\). Plug these values into the Z-score formula to get \(Z = \frac{45 - 65}{15} = -1.33\). This gives us the Z-score for a.
03

Using Z-score table for a.

Using a Z-score table, find the probability that the Z-score is less than -1.33. The value corresponding to Z = -1.33 from the table is approximately 0.0925, which is our answer for a.
04

Using Z-score formula for b.

We have \(x = 79\), \(\mu = 65\), and \(\sigma = 15\). Plugging these values into the Z-score formula gives us \(Z = \frac{79 - 65}{15} = 0.93\). This gives us the Z-score for b.
05

Using Z-score table for b.

Since we want the probability that the Z-score is greater than 0.93, we need to subtract the value corresponding to Z = 0.93 from 1. The value for Z = 0.93 from the table is approximately 0.8238. So, 1- 0.8238 = 0.1762, which is our answer for b.
06

Using Z-score formula for c.

We have \(x = 54\), \(\mu = 65\), and \(\sigma = 15\). Using the Z-score formula gives us \(Z = \frac{54 - 65}{15} = -0.73\). This gives us the Z-score for c.
07

Using Z-score table for c.

To find the probability that the Z-score is greater than -0.73, subtract the value corresponding to Z = -0.73 from 1. The value from the Z = -0.73 from the table is approximately 0.2327. So, 1 - 0.2327 = 0.7673, which is our answer for c.
08

Using Z-score formula for d.

We have \(x = 70\), \(\mu = 65\), and \(\sigma = 15\). The Z-score is given by \(Z = \frac{70 - 65}{15} = 0.33\). This gives us the Z-score for d.
09

Using Z-score table for d.

The probability that Z is less than 0.33 is found directly from the Z-score table. The value corresponding to Z = 0.33 is approximately 0.6293, which is our answer for d.

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